Linear Inequations — MATHS STD 11 Science — Question
Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLinear Inequations1 Mark
Question
Write the solution set of the inequation $|\text{x}-1|\geq|\text{x}-3|$
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Answer
$|\text{x}-1|\geq|\text{x}-3|$ $\Rightarrow|\text{x}-1|-|\text{x}-3|\geq0$ By equating the expression within the modulus to zero, we get x = 1, 3. These point divide real line in three parts viz. $(-\infty,1),[1,3)$ and $[3\infty).$ Case 1: When $-\infty<\text{x}<1$ |x - 1| = -(x - 1) and |x - 3| = -(x - 3) $\therefore|\text{x}-1|-|\text{x}-3|\geq0$ $\Rightarrow-2\geq0$ which is not true. So, the given inequation has no solution for $\text{x}\in(-\infty,1)$ Case 2: When $1\geq\text{x}<3$ |x - 1| = -(x - 1) and |x - 3| = -(x - 3) $\therefore|\text{x}-1|-|\text{x}-3|\geq0$ ⇒ (x - 1) + (x - 3) $\geq$ 0 $\Rightarrow2\text{x}-4\geq0$ $\Rightarrow\text{x}\geq2$ But $1\leq\text{x}\leq3$ Therefore in case the solution set of the given inequation is [2, 3) Case 3: When $3\leq\text{x}<\infty$ |x - 1| = -(x - 1) and |x - 3| = -(x - 3) $\therefore|\text{x}-1|-|\text{x}-3|\geq0$ ⇒ (x - 1) + (x - 3) $\geq$ 0 $\Rightarrow2\geq0$ The solution set of the given inequation is $[3,\infty)$ Combining 1 and 3 we obtain that the solution set of the given inequation is $[2,3)\cup[3,\infty)=[2,\infty)$
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